**Jeremy Gray**

In 1900 David Hilbert went to the second International Congress of
Mathematicians in Paris to give an invited paper. He spoke on |

**Hilbert in 1900**

By 1900 Hilbert had emerged
as the leading mathematician in Germany. He was famous for his
solution of the major problems of invariant theory, and for his great
*Zahlbericht*, or *Report on the theory of numbers*, published
in 1896. In 1899, at Klein's request, Hilbert published *The
foundations of geometry* as part of the commemorations of Gauss and
Weber in Göttingen. Hurwitz saw clearly that the implications of
that little book reached far beyond its immediate field. As he put in
a letter to Hilbert:
*You have opened up an immeasurable field of
mathematical investigation which can be called the "mathematics
of axioms" and which goes far beyond the domain of geometry.*

Hermann Minkowski |
Hilbert was therefore poised to lead the international community of
mathematicians. He consulted with his friends Minkowski and Hurwitz,
and Minkowski advised him to seize the moment, writing: |

One
reason is undoubtedly that, for Hilbert, problem solving and theory
formation went hand in hand. Indeed, some of his problems are not
problems at all, but whole programmes of research: Hilbert's 6th
problem, for example, calls for the axiomatisation of physics. But he
gave good reasons for caring about his problems, and bound it all up
with his inspiring optimism. In opposition to Emil du Bois-Reymond's
fashionable academic pessimism, Hilbert insisted that in mathematics:
*We can know, and we shall know.*

**Hilbert presents his problems**

Hilbert's problems came in
four groups. In the first group were six foundational ones, starting
with an analysis of the real numbers using Cantorian set theory, and
including a call for axioms for arithmetic, and the challenge to
axiomatise physics. The next six drew on his study of (algebraic)
number theory, and culminated with his revival of Kronecker's
*Jugendtraum*, and the third set of six were a mixed bag of
algebraic and geometric problems covering a variety of topics. In the
last group were five problems in analysis - the direction that
Hilbert's own interests were going. He asked for a proof that
suitably smooth elliptic partial differential equations have the type
of solutions that physical intuition (and many a German physics
textbook) suggest, even though it had been known since the 1870s that
the general problem of that kind does not. He made a specific
proposal for advancing the general theory of the calculus of
variations.

The Hilbert problems very quickly succeeded in getting
young mathematicians to create the future that Hilbert had conjectured
- not always accurately, however, as we shall see. The Russian
mathematician Serge Bernstein travelled from Paris to Göttingen
in 1904 to present his proof that, under the conditions Hilbert had
stated, elliptic partial differential equations do have analytic
solutions. It would take more than a single book to describe the
results produced in this mushrooming field in the 20th century. The
opportunistic and power-seeking Bieberbach was another man drawn to
the Hilbert problems for the fame that they could confer. In 1908 he
showed that there are only a finite number of crystallographic groups
in a Euclidean space of any dimension, thus solving part of Hilbert's
18th problem. His results confirmed that there are 17 patterns of
this kind for the Euclidean plane, and 219 patterns (or crystal
structures) for Euclidean 3-dimensional space.

**Hilbert and axiomatisation**

Hilbert himself did not work exclusively on the problems, and nor did most
of his many students, who were instead often drawn in the early 1900s to the
study of `Hilbert space'. After 1909, when his friend Minkowski died,
Hilbert became more and more interested in questions of axiomatisation and
the foundations of mathematics. His interest in axiomatising physics was to
lead to several lecture courses but few publications, and has accordingly
been much misunderstood (see Leo Corry's work for the new picture). Hilbert
lectured regularly and well, and he promoted the view that an axiom was a
fundamental idea from which many others followed. So axioms played a
crucial role in organising theories, as they had done in his *Foundations of
geometry.*

As the years went by, the consistency of arithmetic seemed more of a
challenge. In 1917 Hilbert wrote:
*Since the examination of the consistency is a task that cannot be avoided,
it appears necessary to axiomatise logic itself and to prove that number
theory and set-theory are only parts of logic.*
He imposed high standards on the task. What was needed, he said, were
proofs that:

By 1922 Hilbert became locked into a dispute with Brouwer, who
believed that the human mind was strikingly limited in its ability to
deal with infinite sets. Hilbert's hopes for the future of
mathematics were further darkened by the fact that his best student,
Hermann Weyl, also found Brouwer's ideas attractive, and for a time
seemed willing to forgo certain mathematical arguments on
philosophical grounds. Hilbert attempted to get round their arguments
in 1922 by defining mathematics as a system of signs and
distinguishing carefully between mathematics and meta-mathematics.
Mathematics was to be identified with the stock of provable formulae,
while inference about the content was only admissible at the level of
a new meta-mathematics. On the basis of this distinction between
valid formulae and their interpretations he called for a proof theory
- truly a remarkable idea.

In the event Brouwer withdrew from the
contest, Weyl found that he needed classical analysis for his work on
Lie groups, and the crisis passed, although Hilbert's ideas about
proof theory did not convince the experts. But Hilbert continued,
with his assistants Ackermann and Bernays, and in 1931 he published a
forceful re-statement of his views on the occasion of his becoming an
honorary citizen of his native Königsberg. The lecture ends with
a moving affirmation of his deepest belief about mathematics: *There
are absolutely no unsolvable problems. Instead of the foolish
ignorabimus, our answer is on the contrary: We must know, We shall
know.*

Ironically, the day before Hilbert lectured, the young Austrian
logician Kurt Gödel also lectured in Königsberg on his
incompleteness theorem, the work that is popularly said to have killed
Hilbert's programme, even though, as Gödel said in the famous
paper: *I wish to note expressly that [this theorem does] not
contradict Hilbert's formalistic viewpoint.*

The problem is that
finding such proofs has proved elusive. No agreed place to stand has
been found which compels universal assent (such as the elementary
rules of logic) and which delivers all of set theory. On the other
hand, powerful negative results continued to accumulate. When Alan
Turing showed in 1936 that the decision problem is also unsolvable,
the original hopes for Hilbert's programme were all in tatters.

**The Hilbert problems between the Wars**

The Hilbert problems
themselves have perhaps proved a more enduring part of his legacy than
Hilbert's own work on mathematical logic. Those on number theory have
turned out, given Hilbert's own interests, to have been particularly
well put. Ironically, Hilbert's own formulation of Kronecker's
*Jugendtraum* was misleading, but the Japanese mathematician
Takagi, who had studied under Hilbert in Göttingen in 1900 while
writing his thesis for the University of Tokyo, succeeded with the
crucial generalisation to the abelian case in 1920. In 1923 Emil
Artin bumped into Takagi's work almost by accident and the work he did
as a result enabled Hasse to solve Hilbert's 9th Problem (calling for
a general reciprocity law) in 1927. Hasse had meanwhile also solved
Hilbert's 11th problem, on quadratic forms, in 1923. In 1926 Artin
solved Hilbert's 17th problem in a restricted, but nonetheless very
general, case. Hilbert's 7th Problem (to show that *a ^{b}
* is an
irrational transcendental number when

**The Bourbaki connection**

After the Second World War the
struggle for the heart of mathematics was won by the pure
mathematicians. In this context a vigorous contest developed for the
mantle of Hilbert. Should it go to mathematical logicians, or to
applied mathematicians working in the tradition of Courant and
Hilbert, or to the number theorists and algebraists? The most
powerful advocate of the last of these views, both by word and deed,
was Bourbaki. André Weil and Jean Dieudonné shared a view of
mathematics that put Hilbert centre stage, although it was a Hilbert
created in their own image. They espoused the axiomatic method, which
Dieudonné claimed *has revealed unsuspected analogies and
permitted extended generalizations; the origin of the modern
developments of algebra, topology and group theory is to he found only
in the employment of axiomatic methods.* André Weil made
the connection to Hilbert even more forcefully. In 1947 he quoted
Hilbert: *A branch of science is full of life, as long as it offers
an abundance of problems; a lack of problems is a sign of
death.* Great problems, said Weil, furnish the daily bread on
which the mathematician thrives. And turning to Hilbert's famous list
of problems he singled out the 5th problem (then still unsolved) on
Lie groups, the Riemann hypothesis, and the problem of generalising
the theorems of Kronecker's Jugendtraum which `still escapes us, in
spite of the conjectures of Hilbert himself and the efforts of his
pupils'.

**The Hilbert problems after the War**

Hilbert's Paris address had brilliantly united problems with their
theoretical context and so, for a generation, did Bourbaki. The stock of
the Hilbert problems rose with theirs. The remaining ones acquired an extra
cachet for having held out, and they too began to fall. The 5th problem, on
characterising Lie groups, was solved through the work of Gleason and of
Montgomery and Zippin in 1952. The 14th problem on rings of invariants was
interpreted geometrically by Zariski and then solved in the negative by
Nagata in 1959. In the 1960s and 1970s mathematical logicians turned back
to the Hilbert problems. A high point was reached with the award of a
Fields Medal in 1966 to Paul Cohen for showing that the axiom of choice and
the continuum hypothesis are independent of the other axioms of set theory.
The axiom of choice is not one of Hilbert's problems, but in calling for a
consistent set of axioms for arithmetic, Hilbert had opened the way to
similar analyses of all of mathematics, and indeed by establishing the
independence of the continuum hypothesis, Cohen did indeed settle Hilbert's
1st problem.

In the early 1970s the Russian mathematician Yuri Matijasevich solved
Hilbert's 10th problem (Is there a finite process which determines if a
polynomial equation is solvable in integers?) in the negative, using earlier
papers by the American mathematician Martin Davis, Hilary Putnam, and Julia
Robinson. Julia Robinson had been close to solving this problem herself,
and a powerful collaboration developed between her and Matijasevich, despite
the Cold War. It follows from their work that had Hilbert's 10th problem
been answered positively, Goldbach's conjecture (mentioned by Hilbert in
Paris) would have been answered in the negative - a connection that Hilbert
surely had not suspected.

Few of Hilbert's problems have dwindled with the years. Perhaps the
complete solution of the 5th problem is a case in point. In 1986
Jean-Pierre Serre said:
*Still, it is true that sometimes a theory can be killed. A well-known
example is Hilbert's 5th problem. ... When I was a young topologist, that was
a problem I really wanted to solve - but I could get nowhere. It was
Gleason, and Montgomery-Zippin, who solved it, and their solution all but
killed the problem. What else is there to find in this direction? I can
only think of one question: [but it] seems quite hard - but a solution would
have no application whatsoever, as far as I can see.*

Yet there are numerous examples where the Hilbert touch has proved
beneficial. The great development of algebraic number theory was surely
animated by Hilbert's problems. The topic of partial differential equations
was re-opened by Hilbert with his 19th problem, and much of that rich theory
can be traced back to the work it inspired. Not only are the implications
of the solutions and reformulations of the problems still to be worked out,
some of the problems are still alive. Russian mathematicians have recently
shown that in two cases the original `solutions' were flawed, and have given
different and rigorous accounts. Aspects of both halves of the 16th problem
(on real algebraic curves and on vector fields) are still open questions.
Hilbert's deepest vision, of the intricate dance of theory and problems in
mathematics, is one that all mathematicians share, but few have articulated
as well as he.

The original version of this article, including further photos, appeared in the Newsletter

F. Browder (ed.),

Leo Corry, `David Hilbert and the Axiomatization of Physics (1894-1905)',

Jeremy J. Gray,

*Jeremy Gray is a Senior Lecturer in the Department of Pure Mathematics, The
Open University, UK.*

Hilbert's Problems